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Exact Newton method with third-order convergence to model the dynamics of bubbles in incompressible flow

Abstract

In this letter, we present a computational framework based on the use of the Newton
and level set methods and tailored for the modeling of bubbles with surface tension
in a surrounding Newtonian fluid. We describe a fully implicit and monolithic finite
element method that maintains stability for significantly larger time steps compared
to the usual explicit method and features substantial computational savings. A
suitable transformation avoids the introduction of an additional mixed variable in
the variational problem. An exact tangent problem is derived and the nonlinear
problem is solved by a quadratically convergent Newton method. In addition, we
consider a generalization to the multidimensional case of the Kouâs and McDougallâs
methods, resulting in a faster convergence. The method is benchmarked against
known results with the aim of illustrating its accuracy and robustness.